Prix d'un call sur un zéro coupon dans le cadre du modèle de Vasicek

Rédigé par Yacine - 05 janvier 2013

 

Dans le billet suivant je vais vous parler de l'un des cours que j'ai eu dans le cadre de mes études à l'ENSIIE et sous la tutelle de Monique Jeanblanc.
Nous allons définir ce que sont les modèles de taux d'inétrêt. Nous allons présenter le processus d'Ornstein-Ulhenbeck qui nous sera d'une grande aide lors de l'étude du modèle de Vasicek que nous aborderons juste après. Nous pourrons ainsi calculer le prix d'un zéro coupon et le prix d'un Call sur ce dernier.

Problématique


Dans un modèle Vasicek, quelle est l'espérance conditionnelle et la variance conditionnelle de $\int_t^Tr_sds$ par rapport à $F_t$. Quelle est la dynamique du taux forward. Quel est le prix d’un call de maturité $T$ sur un ZC de maturité $T+1$.
Nous allons donc répondre à ces questions. Pour cela, nous allons dans un premiers temps définir ce que sont les modèles de taux d'inétrêt et définir les notations. Nous allons ensuite présenter le processus d'Ornstein-Ulhenbeck qui nous sera d'une grande aide lors de l'étude du modèle de Vasicek que nous aborderons juste après. Enfin, nous calculerons le prix d'un zéro coupon et le prix d'un Call sur ce dernier.

 

 

Modèles de taux d'intérêt


Les modèles de taux d'intérêt permettent principalement de trouver le "juste prix" et couvrir des obligations et des prêts ou des options sur des obligations et prêts. Souvent, le taux d'intérêt est supposé constant. Dans la réalité, le taux d'intérêt dépend de deux dates : la date d'émission $t$ du prêt et la date d'échéance $T$.

 

En environnement certain

 

Pour un emprunt d'un euro en $t$, la somme $F$ à rembourser en $T$ est : \begin{equation} F(t,T)=e^{(T-t)R(t,T)} \end{equation} $R(t,T)$ étant un taux d'intérêt moyen sur la période de $t$ à $T$.
En l'absence d'opportunité d'arbitrage, et si on suppose connus $(R(t,T))_{t\le T}$ alors $F$ vérifie : \begin{equation} \forall t < u < s , F(t,s)= F(t,u)F(u,s) \end{equation} De plus, il est facile d'admettre que :
\begin{equation} F(t,t) = 1 \end{equation} Les deux relations précédentes entraînent l'existence d'une fonction $r(s)$ telle que : \begin{equation} \forall t < T , F(t,T)= e^{\int_t^T r(s)ds} \end{equation} et par conséquent :
\begin{equation} R(t,T)=\frac{1}{T-t}\int_t^Tr(s)ds \end{equation} La fonction $r(s)$ s'interprète comme le taux d'intérêt instantané.

Définition: Nous appellerons "obligation zéro-coupon" un titre donnant droit à 1 euro en date d'échéance $T$ dont la valeur sera noté $P(t,T)$ à l'instant $t$. Nous avons toujours $P(T,T) =1$ et : \begin{equation} \forall t < T , P(t,T)= e^{-\int_t^Tr(s)ds} \end{equation}

 

 

 

 

En environnement incertain

... Bientôt la suite ...

 

 

Faites virevolter cet article sur internet!

author_avatar

Auteur: Yacine

Ingénieur Financier et Ingénieur Informaticien.

Je suis passionné par la finance et l'économie le jour. Geek invétéré durant les heures les plus sombres, d'où le titre du blog : Le Shadow Blog.

Classé dans : Finance, Mathématiques - Mots clés : Vacisek, zero coupon, option, modèle de taux, Call européen

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